cs.LGSep 10, 2026

Enhancing generalization in endwall film cooling prediction: Incorporating the superposition principle into transformer-based neural operators

Authors: Qineng Wang, Liming Song, Tianyuan Liu, Zhendong Guo

Organizations: Institute of Turbomachinery, Xi’an Jiaotong University, Xi’an, Shaanxi, 710049, People’s Republic of China · ENN Science and Technology Development China Co., Ltd., Langfang, Hebei, 065001, China · Hebei Key Laboratory of Compact Fusion, Langfang, Hebei, 065001, China

Read PDF
Ask AI
Cite

Copy citation

DOINot available for this paper

Abstract

In this study, a physics-enhanced neural operator framework is proposed to enhance the generalization prediction ability of the cooling layout of a turbine endwall with variable number of film holes. Specifically, inspired by the film cooling superposition principle, we propose a film cooling prediction model, namely superposition-based deep neural operator (SDNO), that divides the endwall temperature field prediction into two stages. In the first stage, the cooling layout of a turbine endwall is divided into several sub-parts with randomly assigned film holes, and a Transformer-based neural operator network, namely Calculate Net, is designed to predict the temperature field of each sub-part. Then, in the second stage, another neural operator network, i.e., Super Net, is trained to combine the temperature fields predicted by Calculate Net for each sub-part and obtain the superposed temperature field of the full cooling layout. Additionally, instead of directly taking the film cooling contours as pixel plots, a signed distance function (SDF) which is sensitive to the variable locations of cooling holes, is designed to encode the location information of cooling holes. Furthermore, the proposed endwall film cooling prediction model is trained with the samples that changing the number of film holes from 1-5 with variable locations. Then, the trained prediction shows excellent generalization prediction ability, which can accurately predict the film effectiveness of the cooling layout with 10-20 film cooling holes that are unseen in the training samples. The proposed SDNO also improves prediction accuracy relative to the fully supervised baseline. With the above, the effectiveness of our proposed prediction model has been well demonstrated.

Figures & tables

Figure 1: The Pak-B blade endwall gas film cooling structure diagram
Boundary ConditionSymbolValue
Mainstream outlet Reynolds numberReoutRe_{\text{out}}1.98 ×105\times 10^{5}
Mainstream inlet Mach numberMainMa_{\text{in}}0.029
Mainstream outlet Mach numberMaoutMa_{\text{out}}0.047
Mainstream inlet total temperatureT∞T_{\infty}323K
Mainstream inlet turbulence intensityTuT_{u}6%
Velocity boundary layer thicknessδ\delta33mm
Table 1: Design conditions of the Pak-B blade
Figure 2: Layout mode generates schematic diagram
Figure 3: Non-mixed data set preparation strategy considering the number of film-cooling holes on endwall
Figure 4: The comparison of traditional binary method and SDF method in preprocessing
Network typeOutput size
InputN×128×128×3N\times 128\times 128\times 3
(Linear + RELU) ×\times 5N×16384×64N\times 16384\times 64
Linear attention + residualN×16384×64N\times 16384\times 64
LayerNorm + linear + RELUN×16384×64N\times 16384\times 64
Linear attention + residualN×16384×64N\times 16384\times 64
LayerNorm + linear + RELUN×16384×64N\times 16384\times 64
Table 2: The detailed network structure of Transformer.
Figure 5: The prediction error for the fully supervised baseline network
Figure 6: Using Seller’s formula in the extrapolation prediction of the film-cooling at the endwall of the pak-B blade.
Network typeOutput size
InputN×128×128×2N\times 128\times 128\times 2
Patch16N×32×32×216N\times 32\times 32\times 2
Linear-116N×32×32×6416N\times 32\times 32\times 64
(Conv2d + GELU + SpectralConv2d) ×\times 216N×64×32×3216N\times 64\times 32\times 32
Linear-216N×128×32×3216N\times 128\times 32\times 32
Linear-316N×32×32×116N\times 32\times 32\times 1
Table 3: The detailed network structure of FNO.
Figure 7: The SDNO network structure and training strategy
Figure 8: The sketch map for the decomposition-calculation-superposition steps for a 10-holes sample
Training stepFeed forwardBackward parameterLearning rateSuperposition times
NormalC^(X)\hat{\mathcal{C}}(X)ΘC\Theta_{\mathcal{C}}0.0001—
SuperpositionS^(C^(X(A)),C^(X(B)))\hat{\mathcal{S}}(\hat{\mathcal{C}}(X^{(A)}),\hat{\mathcal{C}}(X^{(B)}))ΘC,ΘS\Theta_{\mathcal{C}},\Theta_{\mathcal{S}}0.000011
Fine-tuningS^(C^(X(A)),C^(X(B)))\hat{\mathcal{S}}(\hat{\mathcal{C}}(X^{(A)}),\hat{\mathcal{C}}(X^{(B)}))ΘS\Theta_{\mathcal{S}}0.000011, 2, 3
Table 4: Forward and backward operations and parameters of the SDNO training strategies.
Figure 9: The prediction error for ID and OOD samples of the computational network
Figure 10: The prediction error of 10-holes samples with different decomposition strategies
Figure 11: The prediction error for ID and OOD samples of the SDNO network
Figure 12: The prediction error for ID and OOD samples of the SDNO network
Figure 13: The comparison of temperature fields from CFD simulation and SDNO prediction.
Figure 14: The prediction error of fine-tuning SDNO network with different training data
Figure 15: The prediction error of fine-tuning SDNO network with different training data
Network1 hole2 holes3 holes5 holes10 holes15 holes20 holes
Baseline0.868%0.352%0.442%0.432%41.884%37.297%33.697%
Pre-train: Calculate only0.773%0.181%0.237%0.464%18.714%16.221%22.686%
Pre-train: Super ×\times 11.001%0.364%0.281%0.447%1.405%5.506%12.564%
Pre-train: Super ×\times 21.416%0.464%0.335%0.481%1.478%3.727%7.311%
Fine-tuning: Super ×\times 11.251%0.445%0.409%0.639%0.942%2.364%11.379%
Fine-tuning: Super ×\times 21.535%0.571%0.511%0.743%1.062%1.391%2.794%
Table 5: Mean relative variance error of the SDNO and baseline models for validation datasets with different numbers of cooling holes. All entries are percentages.
Figure 16: The comparison of temperature fields from CFD simulation and fine-tuned SDNO prediction.
Figure 17: The comparison of axial-average cooling effectiveness from CFD simulation and the prediction models.
Figure 18: The comparison between the simulation and prediction performace with different deep learning networks
More in this topic

Transformer Neural Operator

See all 15 papers

Explore similar work

CardsList

  1. A panoramic aerodynamic performance prediction method for turbomachinery cascades using transformer-enhanced neural operator

    Sep 13, 2026Qineng Wang, Zhendong Guo, Liming Song +1Transformer Neural OperatorPDE Surrogate Modeling

    Institute of Turbomachinery, Xi’an Jiaotong University, Xi’an, 710049, China · ENN Science and Technology Development China Co., Ltd., Langfang, 065001, China · Hebei Key Laboratory of Compact Fusion, Langfang, 065001, China

  2. Temperature Field Reconstruction of Tungsten Monoblock Divertor on EAST using Physics-aware Neural Operator Transformer

    Jun 30, 2026Zikang Yan, Xiao Wang, Qingquan Yang +6Transformer Neural OperatorPDE Surrogate Modeling

    School of Computer Science and Technology, Anhui University, Hefei, China · Institute of Plasma Physics, Chinese Academy of Sciences, Hefei, China · Tongling University, Tongling 244000, China +2

  3. Virtual Temperature Sensors in Power Transformers Using Neural Ordinary Differential Equations

    Aug 13, 2026Berk Hadzhamolla, Alexander Johannes Stasik, Signe Riemer-SørensenPhysics-Informed MLNeural ODEs

    University of Oslo, Oslo, 0316, Norway · 2,3 SINTEF AS, Department of Mathematics and Cybernetics, Oslo, Norway · Department of Data Science, Norwegian University of Life Sciences, ˚As, Norway